somepropertiesoffibonacciandchebyshev polynomials
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Li Advances in Difference Equations (2015) 2015:118 DOI 10.1186/s13662-015-0420-z
R E S E A R C H Open Access
Some properties of Fibonacci and ChebyshevpolynomialsYang Li*
*Correspondence:[email protected] of Mathematics,Northwest University, Xi’an, Shaanxi,P.R. China
AbstractIn this paper, we study the properties of Chebyshev polynomials of the first andsecond kind and those of Fibonacci polynomials and use an elementary method togive Chebyshev polynomials of the first and second kind in terms of Fibonaccipolynomials and vice versa. Finally, we get some identities involving the Fibonaccinumbers and the Lucas numbers.
Keywords: elementary method; Chebyshev polynomials; Fibonacci polynomials;Fibonacci numbers; Lucas numbers
1 IntroductionAs we know, the Chebyshev polynomials and Fibonacci polynomials are usually definedas follows: Chebyshev polynomials of the first kind are Tn+(x) = xTn+(x) – Tn(x), n ≥ ,with the initial values T(x) = , T(x) = x; Chebyshev polynomials of the second kind areUn+(x) = xUn+(x) – Un(x), n ≥ , with the initial values U(x) = , U(x) = x; Fibonaccipolynomials are Fn+(x) = xFn+(x) + Fn(x), n ≥ , with the initial values F(x) = , F(x) = .From the second-order linear recurrence sequences, we have
Tn(x) =[(
x +√
x – )n +
(x –
√x –
)n],
Un(x) =
√
x –
[(x +
√x –
)n+ –(x –
√x –
)n+],
Fn(x) =
n√
x +
[(x +
√x +
)n –(x –
√x +
)n].
These polynomials play a very important role in the study of the theory and applicationof mathematics, and they are closely related to the famous Fibonacci numbers {Fn} andLucas numbers {Ln} which are defined by the second-order linear recurrence sequences
Fn+ = Fn+ + Fn,
Ln+ = Ln+ + Ln,
where n ≥ , F = , F = , L = and L = . Therefore, many authors have investigatedthese polynomials and got many properties and corollaries. For example, Zhang [] uses
© 2015 Li; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, pro-vided the original work is properly credited.
Li Advances in Difference Equations (2015) 2015:118 Page 2 of 12
the Chebyshev polynomials and has obtained the general formulas involving Fn and Ln,
∑
a+a+···+ak+=nFm(a+) · Fm(a+) · · ·Fm(ak++)
= (–i)mn Fk+m
k · k!U (k)
n+k
(imLm
),
∑
a+a+···+ak+=n+k+
Lm(a+) · Lm(a+) · · ·Lm(ak++)
= (–i)m(n+k+) k!
k+∑
h=
(im+Lm
)h (k + )!h!(k + – h)!
U (k)n+k+–h
(imLm
),
where k, m are any positive integers, a, a, . . . , ak+ are nonnegative integers and i is thesquare root of –. Falcón and Plaza [, ] presented many formulas about Fibonacci poly-nomials. This fact allowed them to present a family of integer sequences in a new anddirect way. Zhang [] also used the Chebyshev polynomials to solve some calculating prob-lems of the general summations. Wu and Yang [] studied Chebyshev polynomials and gota lot of properties.
In this paper, we combine Sergio Falcón and Wenpeng Zhang’s ideas. Then we obtainthe following theorems and corollaries. These results strengthen the connections of twokinds of polynomials. They are also helpful in dealing with some calculating problems ofthe general summations or studying some integer sequences.
Theorem For any positive integer n, we have the identities
Tn(x) =n+∑
s=
n∑
k=
k · (sn – n)(n + k – )!Fs–(x)(–)s+n–(n – k)!(k + s)!(k – s + )!
,
Tn–(x) =n∑
s=
n∑
k=
k–(ns – s)(n + k – )!Fs(x)(–)s+n(n – k)!(k + s)!(k – s)!
.
Theorem For any positive integer n, we have
Un(x) =n+∑
s=
n∑
k=
k–( – s)(n + k)!Fs–(x)(–)s+n(n – k)!(k + s)!(k – s + )!
,
Un–(x) =n∑
s=
n–∑
k=
k+s(n + k – )!(–)s+nFs(x)(n – k + )!(k + s + )!(k – s + )!
.
Theorem For any positive integer n, we have the following forms:
Fn–(x) =n–∑
k=
n–∑
j=
j+–n(k + )(n – j – )!j!(n – k – j – )!(n + k – j)!
Uk(x),
Fn(x) =n–∑
k=
n–∑
j=
j–n+k(n – j – )!j!(n – k – j)!(n + k – j)!
Uk(x).
Li Advances in Difference Equations (2015) 2015:118 Page 3 of 12
Theorem For any positive integer n, we have the following forms:
Fn(x) =n∑
k=
n–∑
j=
j+–n(n – j – )!Tk–(x)j!(n + k – j – )!(n – k – j)!
,
Fn–(x) =n–∑
k=
n–∑
j=
j+–n(n – j – )!Tk(x)j!(n + k – j – )!(n – k – j – )!
+n–∑
j=
j+–n(n – j – )!j!(n – j – )!(n – j – )!
.
Corollary For any positive integer n, we have the following identities:
Fn–
(imLm
)=
n–∑
k=
n–∑
j=
j+–n(k + )(n – j – )!Fm(k+)
j!(n – k – j – )!(n + k – j)!(–)kmFm,
Fn
(imLm
)=
n–∑
k=
n–∑
j=
j–n+k(n – j – )!(–i)Fkm
(–)kmj!(n – k – j)!(n + k – j)!Fm.
Corollary For any positive integer n, we have the following identities:
Fn
(imLm
)=
n∑
k=
n–∑
j=
j+–n(n – j – )!(–)km(–i)Lkm–m
j!(n + k – j – )!(n – k – j)!,
Fn–
(imLm
)=
n–∑
k=
n–∑
j=
j+–n(n – j – )!(–)kmLkm
j!(n + k – j – )!(n – k – j – )!
+n–∑
j=
j+–n(n – j – )!j!(n – j – )!(n – j – )!
.
2 Some lemmasLemma For any nonnegative integers m and n, we have these identities
∫
–
Tm(x)Tn(x)√ – x
dx =
⎧⎪⎨
⎪⎩
, m �= n,π , m = n > ,π , m = n = ;
()
∫
–Um(x)Un(x)
√ – x dx =
{, m �= n,π , m = n;
()
Tn(cos θ ) = cos nθ ; ()
Un(cos θ ) =sin (n + )θ
sin θ. ()
Proof See reference []. �
Lemma For any positive integers m and n, we have these identities
Un
(i
)= inFn+,
Tn
(i
)=
in
Ln+,
Li Advances in Difference Equations (2015) 2015:118 Page 4 of 12
Tn
(Tm
(i
))= imnLmn,
Un
(Tm
(i
))= imn Fm(n+)
Fm.
Proof See reference []. �
Lemma For any positive integer n, we have
Tn(Tm(x)
)= Tnm(x),
Un(Tm(x)
)=
Um(n+)–(x)Um–(x)
.
Proof See reference []. �
Lemma For any positive integer n, let
Fn(x) =+∞∑
k=
ankUk(x) ()
and
Fn(x) =
bnT(x) ++∞∑
k=
bnkTk(x), ()
then we can get
ank =
{∑� n– �
j=(k+)(n–j–)!
j!(n–k–j–)!!(n+k+–j)!! , k + n is odd,, otherwise,
()
bnk =
{∑� n– �
j=(n–j–)!
j!(n+k–j–)!!(n–k–j–)!! , k + n is odd,, otherwise.
()
Proof To begin with, we multiply√
– xUm(x) to both sides of (), then integrate it from– to , we can get the following identity by applying property ():
∫
–
√ – xFn(x)Um(x) dx =
∞∑
k=
∫
–ank
√ – xUm(x)Uk(x) dx =
π
anm,
and then we have
anm =π
∫
–
√ – xFn(x)Um(x) dx.
From reference [] we know
Fn(x) =� n–
�∑
j=
(n – j –
j
)xn–j–, ()
Li Advances in Difference Equations (2015) 2015:118 Page 5 of 12
where n ≥ . We define
wnk =π
∫ π
cosn θ sin(k + )θ sin θ dθ .
From reference [] we know
wnk =
{(k+)n!
(n+k+)!!(n–k)!! , k + n is even and n ≥ k,, otherwise,
()
where n and k are any nonnegative integers. Let x = cos θ , then we can get the followingidentity by applying property () and property ():
anm =π
∫ π
Fn(cos θ ) sin(m + )θ sin θ dθ
=π
� n– �∑
j=
(n – j –
j
)∫ π
cosn–j– θ sin(m + )θ sin θ dθ
=� n–
�∑
j=
(n – j –
j
)wn–j–,m,
and then we have anm = if n + m is even. If n + m is odd, we have
anm =� n–
�∑
j=
(n – j –
j
)· (m + )(n – j – )!
(n – m – j – )!!(n + m + – j)!!
=� n–
�∑
j=
(n – j – )!j!(n – j – )!
· (m + )(n – j – )!(n – m – j – )!!(n + m + – j)!!
=� n–
�∑
j=
(m + )(n – j – )!j!(n – m – j – )!!(n + m + – j)!!
.
We finish proving property ().In order to prove property (), we must multiply Tm(x)√
–x to both sides of (), then integrateit from – to , we can get the following identity by applying property ():
∫
–
Fn(x)Tm(x)√ – x
dx =∫
–
bnT(x)Tm(x)√
– xdx +
∞∑
k=
∫
–
bnkTm(x)Tn(x)√ – x
dx =π
bnm,
and then we have
bnm =π
∫
–
Fn(x)Tm(x)√ – x
dx.
We define
qnk =π
∫ π
cosn θ cos kθ dθ .
Li Advances in Difference Equations (2015) 2015:118 Page 6 of 12
From reference [] we know
qnk =
{n!
(n+k)!!(n–k)!! , k + n is even and n ≥ k,, otherwise,
()
where n and k are any nonnegative integers. Let x = cos θ , then we can get the followingidentity by applying property () and property ():
bnm =π
∫ π
Tm(cos θ )Fn(cos θ )sin θ
sin θ dθ
=π
� n– �∑
j=
(n – j –
j
)∫ π
cosn–j– θ cos mθ dθ
=� n–
�∑
j=
(n – j –
j
)qn–j–,m,
and then we have bnm = if n + m is even. If n + m is odd, we have
bnm =� n–
�∑
j=
(n – j –
j
)· (n – j – )!
(n + m – j – )!!(n – m – j – )!!
=� n–
�∑
j=
(n – j – )!j!(n – j – )!
· (n – j – )!(n + m – j – )!!(n – m – j – )!!
=� n–
�∑
j=
(n – j – )!j!(n + m – j – )!!(n – m – j – )!!
.
This proves Lemma . �
Lemma For any positive integers m and n, we have the following identities:
Fn(i cos θ ) =in+ sin nθ
sin θ, ()
∫ i
–i
√x + Fm(x)Fn(x) dx =
{im–π , m = n > ,, otherwise.
()
Proof As we know,
Fn(x) =
n√
x +
[(x +
√x +
)n –(x –
√x +
)n].
Let x = i cos θ , then we have
Fn(i cos θ ) =
sin θ
[(i cos θ + sin θ )n – (i cos θ – sin θ )n]
=
sin θ
(ine–inθ – ineinθ
)
Li Advances in Difference Equations (2015) 2015:118 Page 7 of 12
=in
sin θ(cos nθ – i sin nθ – cos nθ – i sin nθ )
=in+ sin nθ
sin θ.
This proves property (). Let x = i cos θ in the following identity:
A =∫ i
–i
√x + Fm(x)Fn(x) dx,
then we can get
A =∫ π
sin θFn(i cos θ )Fm(i cos θ )i sin θ dθ
=∫ π
i sin θ
in+ sin nθ
sin θ
im+ sin mθ
sin θdθ
= in+m–∫ π
sin nθ sin mθ dθ
= in+m–∫ π
cos (n – m)θ – cos (m + n)θ dθ .
Then we can get property (). This proves Lemma . �
Lemma For any positive integer n, we have
Tn(x) =n∑
k=
(–)n–k · k · nn + k
(n + k
k
)xk ,
Tn+(x) =n∑
k=
(–)n–k · k · (n + )n + k +
(n + k +
k +
)xk+,
Un(x) =n∑
k=
(–)n–k · k · (k + )n + k +
(n + k +
k +
)xk ,
Un+(x) =n∑
k=
(–)n–k · k+ · (k + )n + k +
(n + k +
k +
)xk+.
Proof From Theorem of reference [], we can get the following result easily:
Tn(x) =n∑
k=
(–)n–k[
k(
n + kn – k
)– k–
(n + k –
k –
)]xk
=n∑
k=
(–)n–k · k–[
(n + k)!(k)!(n – k)!
–(n + k – )!
(k – )!(n – k)!
]xk
=n∑
k=
(–)n–k · k n(n + k – )!(k)!(n – k)!
xk
=n∑
k=
(–)n–k · k · nn + k
(n + k
k
)xk .
Li Advances in Difference Equations (2015) 2015:118 Page 8 of 12
From Theorem of reference [], we know
Tn+(x) =n∑
k=
(–)n–k[
k+(
n + k + n – k
)– k
(n + k
k
)]xk+.
In a similar way, we can get
Tn+(x) =n∑
k=
(–)n–k · k · (n + )n + k +
(n + k +
k +
)xk+.
We can get T ′n(x) = nUn–(x) easily from the definition of the Chebyshev polynomials. If
we derive both sides of the above properties, we will get
Un(x) =n∑
k=
(–)n–k · k · (k + )n + k +
(n + k +
k +
)xk ,
Un+(x) =n∑
k=
(–)n–k · k+ · (k + )n + k +
(n + k +
k +
)xk+.
This proves Lemma . �
Lemma For any positive integer n, let
Tn(x) =+∞∑
s=
cn,sFs(x) ()
and
Tn–(x) =+∞∑
s=
cn–,sFs(x), ()
then we can get
cn,s =
{∑nk=
k+sn·is+n+(n+k–)!(n–k)!(k+s+)!!(k–s+)!! , s is odd,
, otherwise,()
cn–,s =
{∑nk=
(ns–s)is+n(n+k–)!–k (n–k)!(k+s)!!(k–s)!! , s is even,
, otherwise.()
Proof At first, we multiply√
x + Fm(x) to both sides of (), then integrate it from –ito i. We can get the following identity by applying Lemma , where m is any positiveinteger:
∫ i
–i
√x + Fm(x)Tn(x) dx =
∞∑
s=
∫ i
–icn,s
√x + Fs(x)Fm(x) dx = im–πcn,m,
and then we have
cn,m =(–i)m–
π
∫ i
–i
√x + Fm(x)Tn(x) dx.
Li Advances in Difference Equations (2015) 2015:118 Page 9 of 12
Let x = i cos θ , then we can get the following identity by applying Lemma and property():
cn,m =(–i)m–
π
∫ π
Tn(i cos θ )Fm(i cos θ )i sin θ dθ
=–im
π
∫ π
Tn(i cos θ ) · im+ sin mθ
sin θ· ( sin θ ) dθ
=im+
π
∫ π
Tn(i cos θ ) sin mθ sin θ dθ
=n∑
k=
k · n · im+n+
n + k
(n + k
k
)wk,m–,
so when m is even, we have cn,m = . When m is odd, we have
cn,m =n∑
k=
k · n · im+n+
n + k·(
n + kk
)· m · (k)!
(k + m + )!!(k – m + )!!
=n∑
k=
k · n · im+n+
n + k· (n + k)!
(k)!(n – k)!· m · (k)!
(k + m + )!!(k – m + )!!
=n∑
k=
k+ · mn · im+n+(n + k – )!(n – k)!(k + m + )!!(k – m + )!!
.
In a similar way, we can get the following result easily:
cn–,m =(–i)m–
π
∫ π
Tn–(i cos θ )Fm(i cos θ )i sin θ dθ
=n∑
k=
k– · (n – ) · im+n
n + k –
(n + k –
k –
)wk–,m–,
and we have cn–,m = if m is odd. If m is even, we have
cn–,m =n∑
k=
k– · (n – ) · im+n
n + k –
(n + k –
k –
)m(k – )!
(k + m)!!(k – m)!!
=n∑
k=
(nm – m)im+n(n + k – )!–k(n – k)!(k + m)!!(k – m)!!
.
This proves Lemma . �
Lemma For any positive integer n, let
Un(x) =+∞∑
s=
dn,sFs(x) ()
and
Un–(x) =+∞∑
s=
dn–,sFs(x), ()
Li Advances in Difference Equations (2015) 2015:118 Page 10 of 12
then we can get
dn,s =
{∑nk=
k+·is+n+s(n+k)!(n–k)!(k+s+)!!(k–s+)!! , s is odd,
, otherwise,()
dn–,s =
{∑n–k=
k+·is+ns(n+k–)!(n–k–)!(k+s+)!!(k+–s)!! , s is even,
, otherwise.()
Proof At first, we multiply√
x + Fm(x) to both sides of (), then integrate it from –ito i, we can get the following identities by applying Lemma , where m is any positiveinteger:
∫ i
–i
√x + Fm(x)Un(x) dx =
∞∑
k=
∫ i
–idns
√x + Fs(x)Fm(x) dx = im–πdn,m,
then we have
dn,m =(–i)m–
π
∫ i
–i
√x + Fm(x)Un(x) dx.
Let x = i cos θ , then we can get the following identity by applying Lemma and property():
dn,m =im+
π
∫ π
Un(i cos θ )Fm(i cos θ )i sin θ dθ
=im+
π
∫ π
Un(i cos θ )
im+ sin mθ
sin θ( sin θ ) dθ
=im+
π
∫ π
Un(i cos θ ) sin mθ sin θ dθ
=n∑
k=
k · (k + ) · im++n
n + k +
(n + k +
k +
)wk,m–,
so we have dn,m = if m is even. If m is odd, we have
dn,m =n∑
k=
k · (k + ) · im++n
n + k +
(n + k +
k +
)· m · (k)!
(k + m + )!!(k – m + )!!
=n∑
k=
k · (k + ) · im++n
n + k + · (n + k + )!
(k + )!(n – k)!· m · (k)!
(k + m + )!!(k – m + )!!
=n∑
k=
k+ · im+n+m(n + k)!(n – k)!(k + m + )!!(k – m + )!!
.
In a similar way, we have
dn–,m =(–i)m–
π
∫ π
Un–(i cos θ )Fm(i cos θ )i sin θ dθ
=n–∑
k=
k+ · (k + ) · im+n
n + k +
(n + k + k +
)wk+,m–.
Li Advances in Difference Equations (2015) 2015:118 Page 11 of 12
When m is odd, we have dn–,m = . When m is even, we have
dn–,m =n–∑
k=
k+ · (k + ) · im+n
n + k +
(n + k + k +
)m(k + )!
(k + m + )!!(k + – m)!!
=n–∑
k=
k+ · im+nm(n + k – )!(n – k – )!(k + m + )!!(k + – m)!!
.
This proves Lemma . �
3 Proof of the theorems and corollariesIn this section, we will prove our theorems and corollaries. First of all, we can prove all thetheorems from Lemma , Lemma and Lemma easily.
Proof of Corollary We can get the following properties from Lemma and Lemma byletting x = Tm(x) in Theorem :
Fn–(Tm(x)
)=
n–∑
k=
n–∑
j=
j+–n(k + )(n – j – )!j!(n – k – j – )!(n + k – j)!
Uk(Tm(x)
),
Fn(Tm(x)
)=
n–∑
k=
n–∑
j=
j–n+k(n – j – )!j!(n – k – j)!(n + k – j)!
Uk(Tm(x)
).
Then, taking x = i in the above identities, according to Lemma , we can get Corollary .
�
Proof of Corollary We can get the following properties from Lemma and Lemma byletting x = Tm(x) in Theorem :
Fn(Tm(x)
)=
n∑
k=
n–∑
j=
j+–n(n – j – )!Tk–(Tm(x))j!(n + k – j – )!(n – k – j)!
,
Fn–(Tm(x)
)=
n–∑
k=
n–∑
j=
j+–n(n – j – )!Tk(Tm(x))j!(n + k – j – )!(n – k – j – )!
+n–∑
j=
j+–n(n – j – )!j!(n – j – )!(n – j – )!
T(Tm(x)
).
Then, taking x = i in the above identities, according to Lemma , we can get Corollary .
�
Competing interestsThe author declares that there is no conflict of interests regarding the publication of the article.
AcknowledgementsThe author would like to thank the referees for their very helpful and detailed comments which have significantlyimproved the presentation of this paper.
Received: 28 November 2014 Accepted: 18 February 2015
Li Advances in Difference Equations (2015) 2015:118 Page 12 of 12
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